Understanding Polygons

A polygon is a 2D shape enclosed by three or more line segments. These straight line segments are called edges or sides of the polygon. Examples of polygons include triangle, square, quadrilateral, pentagon, hexagon etc.

The common endpoint of two adjoining sides of the polygon is called the vertex of the polygon.

A line segment joining the non-adjacent vertices of a polygon is called the diagonal of the polygon.

Polygons can be regular or irregular.

Regular Polygons

In a regular polygon all angles are equal and all sides have the same length.

Interior and Exterior angles

A polygon has interior and exterior angles. The interior angle of a polygon is the angle within the vertex. (angles a, b, c, d, e in the figure below). The exterior angle is the angle formed by one side of the polygon and the line extended from the adjacent side (angle f in the figure below). Thus, the corresponding interior and the exterior angles lie on a straight line (angles a and f respectively).

Therefore, at each vertex of a polygon:

Interior angle + Exterior angle = 180° (Linear pair of angles)

The exterior angles of a polygon add up to 360°

If each angle of a polygon is less than 180°, then it is called a convex polygon

If atleast one of the angles of a polygon is more than 180°, then it is called a concave polygon

Practice Questions

Q1: Find the measure of the interior angles of a regular pentagon

Answer:

You can find the sum of the interior angles by dividing the pentagon into triangles. A pentagon can be divided into three triangles.

Therefore, sum of the interior angles of a pentagon

= 3 x 180° (Sum of angles of a triangle = 180°)

= 540°

To find the interior angle, divide the sum of all interior angles by the number of sides

= 540 / 5

= 108°

The measure of the interior angle in a regular pentagon is 108°.

Q2: Find the measure of the exterior angle of a regular hexagon. Also find measure of its interior angles.

Answer:

The sum of the exterior angles of a polygon is 360°. Since a hexagon has 6 sides, the exterior angle can be calculated as

= 360 / 6

= 60°

The exterior angle of a regular hexagon is 60°

In a polygon, the sum of the exterior and the adjacent interior angles is 180° (Linear pair of angles)

Therefore, interior angle = 180 - 60 = 120°

The interior angle of a regular hexagon is 120°.

Q3: Find the number of sides of a regular polygon given that the measure of each interior angle of 120°.

Answer:

For a regular polygon, the sum of the exterior angle and adjacent interior angle is 180°.

Therefore, exterior angle = 180 - 120 = 60°

The sum of the exterior angles of a regular polygon is 360°

Therefore, number of exterior angles

= 360 / 60

= 6

There are 6 exterior angles and therefore there are six sides in this polygon. Hence, this polygon is a hexagon.

Q4: Find the number of sides of a regular polygon given that the measure of each interior angle of 108°.

Answer:

For a regular polygon, The sum of the exterior angle and adjacent interior angle is 180°.

Therefore, exterior angle = 180 - 108 = 72°

The sum of the exterior angles of a regular polygon is 360°

Therefore, number of exterior angles

= 360 / 72

= 5

There are 5 exterior angles and therefore there are 5 sides in this polygon. Hence, this polygon is a pentagon.

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